theo mx+y=1 =>y=1-mx

ta có x-my=2=>x-m(1-mx)=2=>x-m+m^2x=2=>x(1+m^2)=2+m=>x=(2+m)/(1+m^2)

thay vô tim y=(2m^2-m)/((1+m^2)/m)

thay vô x+y=1 tim ra m

\(\hept{\begin{cases}3x-y=2m+3\\x+2y=3m+1\end{cases}}\Leftrightarrow\hept{\begin{cases}6x-2y=4m+6\\x+2y=3m+1\end{cases}}\)

\(\Leftrightarrow\hept{\begin{cases}x=m+1\\y=m\end{cases}}\)khi đó: \(^{x^2+y^2=5\Leftrightarrow2m^2+2m+1=5\Leftrightarrow2m^2+2m-4=0\Leftrightarrow\orbr{\begin{cases}m=1\\m=-2\end{cases}}}\)

\(=3x^4+3x^2y^2+2x^2y^2+2y^4+2y^2\)

\(=\left(3x^2+2y^2\right)\left(x^2+y^2\right)+2y^2\)

\(=3x^2+2y^2+2y^2\)

\(=3x^2+4y^2\)

Lấy (1) cộng (2), ta có:

\(\left(2a+1\right)x=a^2+4a+5\)\(\Rightarrow x=\dfrac{a^2+4a+5}{2a+1}\)

Thay vào (1): \(\dfrac{\left(a^2+4a+5\right)\left(a+1\right)-10a-5}{2a+1}.\dfrac{1}{a}\)\(=\dfrac{a^3+5a^2-a}{2a+1}.\dfrac{1}{a}=\dfrac{a^2+5a-1}{2a+1}\)

Để x,y nguyên thì \(\left\{{}\begin{matrix}a^2+4a+5⋮2a+1\\a^2+5a-1⋮2a+1\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}a\left(a+2\right)+2a+5⋮2a+1\\a^2+2a+3a-1⋮2a+1\end{matrix}\right.\)\(\Rightarrow\left\{{}\begin{matrix}4⋮2a+1\\a+2⋮2a+1\end{matrix}\right.\)\(\Rightarrow\left\{{}\begin{matrix}4⋮2a+1\\3⋮2a+1\end{matrix}\right.\)\(\Rightarrow2a+1\in\left\{\pm1\right\}\)\(\Rightarrow a\in\left\{-1;0\right\}\)

Vậy với a=-1;0 thì hpt có nghiệm (x;y) với x,y thuộc Z.